{
  "abstract": "Lazard theory is becoming increasingly important in $p$-adic representation theory, especially when studying mod $p$ representations. For $G$ a connected reductive group defined over a finite extension $K/\\mathbb Q_p$, a question going back to Lazard is when a pro-$p$ Iwahori subgroup of $G(K)$ admits a saturated $p$-valuation. In this article, we establish a deep connection to Moy-Prasad theory, and show that the Moy-Prasad filtration defines a saturated $p$-valuation on $G_{x,\\frac{1}{p-1}+}$ for any point $x$ in the semisimple Bruhat-Tits building of $G$, given a mild assumption on the maximal torus of $G$ and that $p>2$. We also compute Lazard Lie algebras defined over $\\mathbb Z_p$ and the polynomial ring $\\mathbb F_p[\\varepsilon]$ associated to the saturated group $G_{x,\\frac{1}{p-1}+}$, which allows us to reinterpret the Moy-Prasad isomorphism as an isomorphism of Lie algebras over $\\mathbb F_p[\\varepsilon]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13877",
  "author_total": 1,
  "authors": [
    {
      "name": "Casey Perdue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7g0tfsaaj2n4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13877",
  "fulltext_url": null,
  "id": "a7g0tfsaaj2n4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13877",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221524066",
  "pdf_url": "https://arxiv.org/pdf/2610.13877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7g0tfsaaj2n4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the saturability of $p$-adic Lie groups",
  "title_text": "On the saturability of p-adic Lie groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/a7g0tfsaaj2n4",
  "venue": "arXiv",
  "versions": []
}